On $L^{3,\infty}$-stability of the Navier-Stokes system in exterior domains

Abstract: This paper studies the stability of a stationary solution of the Navier-Stokes system with a constant velocity at infinity in an exterior domain. More precisely, this paper considers the stability of the Navier-Stokes system governing the stationary solution which belongs to the weak $L^3$-space $L^{3,\infty}$. Under the condition that the initial datum belongs to a solenoidal $L^{3 , \infty}$-space, we prove that if both the $L^{3,\infty}$-norm of the initial datum and the $L^{3,\infty}$-norm of the stationary solution are sufficiently small then the system admits a unique global-in-time strong $L^{3,\infty}$-solution satisfying both $L^{3,\infty}$-asymptotic stability and $L^\infty$-asymptotic stability. Moreover, we investigate $L^{3,r}$-asymptotic stability of the global-in-time solution. Using $L^p$-$L^q$ type estimates for the Oseen semigroup and applying an equivalent norm on the Lorentz space are key ideas to establish both the existence of a unique global-in-time strong (or mild) solution of our system and the stability of our solution.